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International Journal of Heat and Mass Transfer 50 (2007) 4039–4051

CHF model for subcooled flow boiling in Earth gravityand microgravity

Hui Zhang a, Issam Mudawar a,*, Mohammad M. Hasan b

a Boiling and Two-phase Flow Laboratory, School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, USAb NASA Glenn Research Center, 21000 Brookpark Road, Cleveland, OH 44135, USA

Received 8 September 2006; received in revised form 26 January 2007Available online 27 March 2007

Abstract

This study is the first attempt at extending the Interfacial Lift-off CHF Model to subcooled flow boiling conditions. A new CHF data-base was generated for FC-72 from ground tests as well as from microgravity tests that were performed in parabolic flight trajectory.These tests also included high-speed video imaging and analysis of the liquid–vapor interface during the CHF transient. Both theCHF data and the video records played a vital role in constructing and validating the extended CHF model. The fundamental differencebetween the original Interfacial Lift-off Model, which was developed for saturated flow boiling, and the newly extended model is thepartitioning of wall energy between sensible and latent heat for subcooled flow boiling. This partitioning is modeled with the aid of anew ‘‘heat utility ratio”. Using this ratio, the extended Interfacial Lift-off Model is shown to effectively predict both saturated and sub-cooled flow boiling CHF in Earth gravity and in microgravity.� 2007 Published by Elsevier Ltd.

1. Introduction

Subcooled flow boiling is a highly efficient means ofremoving heat from high-heat-flux dissipating surfaces. Itis used in cooling nuclear reactor cores and absorbing heatfrom combustion products in power plant boilers. In recentyears, thermal management needs in the electronics andaerospace industries have spurred unprecedented interestin implementing flow boiling to capitalize upon its poten-tial to produce large heat transfer coefficients.

This important attribute has also been a primary reasonfor recent intense efforts to utilize flow boiling in a broadvariety of future space systems and subsystems, includingRankine cycle power generation, electronic cooling, lifesupport, and waste management. With the significantenhancement in heat transfer coefficient compared tosingle-phase cooling, flow boiling is expected to yield

0017-9310/$ - see front matter � 2007 Published by Elsevier Ltd.

doi:10.1016/j.ijheatmasstransfer.2007.01.029

* Corresponding author. Tel.: +1 765 494 5705; fax: +1 765 494 0539.E-mail address: [email protected] (I. Mudawar).

order-of-magnitude reduction in weight-to-heat-load ratio,which is a primary goal in the design of space systems.

Like most boiling systems, heat removal in space is lim-ited by the critical heat flux (CHF). Exceeding this limitcauses a substantial decrease in the heat transfer coefficient,which, for heat-flux controlled surfaces, can trigger seriousphysical damage to the surface being cooled. AccurateCHF prediction is therefore crucial to the design and safetyanalysis of flow boiling systems.

Flow boiling CHF has been explored extensively sincethe 1940s. For the most part, CHF determination has reliedon empirical correlations. Those correlations are valid forthe specific geometries and operating conditions of theexperiments upon which they are based. Due to the depen-dence of CHF on a relatively large number of parameters(geometry, pressure, flow rate, subcooling, etc.), and giventhe limited range of parameters for most correlations, it isoften quite difficult for a designer of a new thermal deviceto find a correlation that covers the entire range of all rel-evant parameters. Correlations are therefore sometimesapplied beyond their range of validity. Such extrapolations

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Nomenclature

A cross-sectional area of channelAk cross-sectional area occupied by phase kAw area of wetting frontb ratio of wetting front length to wavelength, w/kc wave speedCf,i interfacial friction factorci imaginary component of wave speedcp,f specific heat of liquidcr real component of wave speedDk hydraulic diameter for phase kfk wall friction factor for phase k

G mass flux, qf Uga component of acceleration acting axially oppo-

site to direct of fluid flowge earth’s gravitational accelerationgn component of acceleration normal to heated

wallH channel height (5.0 mm)hb enthalpy of bulk liquidhg enthalpy of saturated vaporhfg latent heat of vaporizationhi enthalpy of liquid at inletDhsub enthalpy difference between saturated liquid and

bulk liquidk wave number, 2p=kkc critical wave number, 2p=kc

L heater length in flow direction (101.6 mm)P pressureph heated perimeterpi interfacial perimeter between phasespk perimeter of wall contact with phase k

Po outlet pressureq00 wall heat fluxq00m critical heat fluxq00w wetting front lift-off heat fluxReD,k Reynolds number for phase k based on phase

hydraulic diametert timeT temperatureDT sub;o outlet subcooling, T sat;o � T b;o

U mean liquid inlet velocity

Uf mean liquid phase velocityUg mean vapor phase velocityUg,n mean vapor velocity in wetting front normal to

wallW heater and channel width (2.5 mm)W 0

fg rate of interfacial evaporation per unit stream-wise distance

x flow qualityz streamwise coordinatez0 streamwise distance where U f ¼ Ug

z* extent of continuous upstream wetting region

Greek symbols

a void fraction, d=Hd mean vapor layer thickness; vapor layer ampli-

tude used in CHF modelg interfacial perturbationg0 amplitude of interfacial perturbation, g0 ¼ dk vapor wavelengthkc critical wavelengthkj wavelength of jth wavelk phase viscosityn heat utility ratioqf modified liquid densityqg modified vapor densityqk density of saturated phase kr surface tensionsi interfacial shear stresssw,k shear stress between wall and phase k

Subscripts

b bulk liquidf saturated liquidg saturated vapori inlet; imaginary componentk Phase k (k = f for liquid or g for vapor)m maximum, critical heat fluxo outletsat saturationw wetting front; wall

4040 H. Zhang et al. / International Journal of Heat and Mass Transfer 50 (2007) 4039–4051

may be quite questionable, given the great safety concernsassociated with CHF determination.

To alleviate extrapolation concerns, a smaller number ofstudies have been focused on physical understanding of theCHF phenomenon in pursuit of a mechanistic theoreticalmodel that may be applied to different fluids and broadranges of the relevant parameters. Unfortunately, this is avery complex endeavor, given the limited number of photo-graphic studies that capture the near-wall interfacial behav-ior that is vital to the development of such a model. Severalmechanisms have been proposed to construct such models.

They include Boundary Layer Separation [1,2], BubbleCrowding [3–5], Sublayer Dryout [6–12], and Interfacial

Lift-off [13–20].Very few studies have been dedicated to flow boiling in

microgravity. The majority of these concern bubble behav-ior and two-phase heat transfer coefficient [21–25]. Micro-gravity flow boiling CHF data are especially rare. Ma andChung [24] attempted to obtain such data in a 2.1 s droptower. They generated boiling curves spanning the single-phase and nucleate boiling regions including the CHFpoint for three flow velocities. The authors of the present

Fig. 1. (a) Flow channel assembly and (b) construction of heated wall.

H. Zhang et al. / International Journal of Heat and Mass Transfer 50 (2007) 4039–4051 4041

study conducted near-saturated flow boiling CHF experi-ments onboard NASA’s KC-135 turbojet [20]. CHF datawere measured for near-saturated conditions and flowvelocities from 0.1 to 1.5 m/s. CHF values were unusuallysmall at low velocities in lge compared to 1ge, but the effectof gravity diminished greatly at high velocities. Zhang et al.also conducted extensive high-speed video imaging studiesof flow boiling behavior that proved flow boiling CHF inmicrogravity is triggered by the Interfacial Lift-offmechanism.

A key limitation of the Interfacial Lift-off Model is itsinability to tackle subcooled flows. The present study isthe first attempt at extending this model to subcooled con-ditions. New CHF data were measured in both 1 ge groundexperiments and lge parabolic flight experiments. A newrelation is derived to estimate the partitioning of wall heatflux between sensible and latent heat. This relation is incor-porated in the original Interfacial Lift-off Model to predictdata for both gravitational fields corresponding to differentflow velocities and subcoolings.

2. Experimental methods

2.1. Flow boiling module

A flow boiling test module was designed to both mea-sure flow boiling CHF and photographically capture inter-facial behavior. The module consisted of two transparentpolycarbonate plastic (Lexan) plates and a heater assem-bly. As shown in Fig. 1a, a 5.0 � 2.5 mm rectangular flowchannel was milled into the underside of the top plasticplate. The heated wall consisted of a 0.56 mm thick and101.6 mm long copper plate that was heated by a seriesof thick film resistors. The copper plate was sandwichedbetween the two plastic plates. A honeycomb flow straight-ener at the channel inlet and an entry length 106 times thechannel hydraulic diameter ensured fully developed flow atthe upstream edge of the heated wall. Two thermocouplesprovided fluid temperature readings just upstream anddownstream of the heated wall with an uncertainty of0.3 �C. Pressure transducers at the same locations providedpressure readings with an accuracy of 0.01%.

The heater assembly was designed to provide uniformheat flux along the surface of the copper plate. As shownin Fig. 1b, the heater assembly was formed by solderingsix thick-film resistors to the underside of the copper plate.The resistors were connected in parallel to a variable volt-age transformer. The heater assembly featured fast temper-ature response to changes in heat flux and gravitationalacceleration, which is crucial for flight experiments. Withthe small thickness of the copper plate and small thermalmass of the resistors, the wall temperature could reachsteady state in less than 5 s following a heat flux increment.The 0.56 mm thickness of the copper plate was sufficientlylarge to preclude any CHF dependence on copper platethickness. A detailed discussion on the heater’s tempera-ture response is available in [20]. Wall temperature was

measured by five thermocouples inserted in the copperplate. These thermocouples had an uncertainty of 0.3 �C.The heat flux was determined by dividing the electricalpower input to the resistors by the wetted area of theheated wall. Power input was measured by a power meter,and the overall uncertainty of the heat flux measurementwas 0.2 W/cm2.

2.2. Flow loop

Fig. 2a shows a schematic diagram of the test loop. Thiscompact two-phase flow loop was used to both deaeratethe working fluid, FC-72, prior to testing and maintaindesired flow conditions during the tests. Past experimentsand repeatability tests by the authors have shown deaerat-ing the fluid for about 30 min ensured the removal of anynon-condensible gases from the loop. The fluid was circu-lated in the loop with the aid of a centrifugal pump. Flowrate was controlled by a throttling valve situated down-stream of the pump. The fluid then passed through a filter,a turbine flow meter, and an in-line electric heater, beforeentering the flow boiling module. Exiting the module, thefluid passed through an air-cooled heat exchanger beforereturning to the pump. Fluid temperature was regulatedby controlling the heat exchanger’s fan speed and fine-tuned with the aid of the in-line electric heater. Fluid pres-sure downstream of the heated wall was maintained withthe aid of an accumulator charged with nitrogen gas. Asdepicted in Fig. 2b, the entire test facility, including the

Fig. 2. (a) Schematic of two-phase flow loop and (b) photo of flight apparatus.


flow loop components, power and instrumentation cabi-nets, and data acquisition system, was mounted onto arigid extruded aluminum frame.

Experiments were performed in microgravity and laterrepeated at 1 ge to validate the CHF model for bothenvironments.

2.3. Parabolic flight

The reduced gravity environment was simulated aboarda plane that flew a series of parabolic maneuvers as illus-trated in Fig. 3a. Reduced gravity conditions such asmicrogravity, Lunar gravity (0.16 ge) and Martian gravity

Fig. 3. (a) Trajectory and (b) gravitational change of parabolic flight.


(0.38 ge), were achieved with different parabolic maneuvers.Fig. 3b shows the variation of gravitational accelerationrecorded during microgravity experiments. During amicrogravity parabola, a 23 s microgravity period wasachieved between two 20 s 1.8 ge periods. A flight missionusually consisted of 4 sets of 10 parabolas with about5 min break between consecutive sets. Ten flight missions,about 450 parabolas, were dedicated to the present studyaboard two different aircraft, NASA’s KC-135 turbojetand Zero-G Corporation’s Boeing 727–200.

During a parabolic flight experiment, the desired fluidflow conditions were set before each set of parabolas.Power to the heater assembly was set and maintainedbefore pulling out from a 23 s microgravity period. The

Fig. 4. CHF transient along downstream section of heated wa

power was then increased by an increment of 1–3 W/cm2

until a sudden unsteady rise in wall temperature wasdetected during a microgravity period. Flow boiling CHFin microgravity is defined as the last heat flux measuredprior to the unsteady temperature rise. Besides measuringCHF, vapor behavior was monitored near the outlet ofthe heated wall with the aid of a high-speed video camera.

3. CHF model

The present CHF model is an extension of the Interfa-cial Lift-off Model first conceived by Galloway and Muda-war [13,14] for near-saturated conditions and short heatedwalls and later modified by Gersey and Mudawar [15,16]for longer walls. Later, Sturgis and Mudawar [17,18]observed that CHF in subcooled flow boiling was also trig-gered by interfacial lift-off but did not formulate a modelfor these conditions. They observed a periodic distributionof vapor patches along the heated surface immediatelyprior to CHF. Liquid replenishment of the near-wall regionand cooling of the surface occurred only at discrete wettingfronts between the vapor patches. Severe vapor effusion ina wetting front lifted the vapor-liquid interface away fromthe surface. The formation of a continuous vapor blanketensued and heat transfer in the remaining wetting frontsincreased in order to compensate for the loss of the liftedwetting front. This caused accelerated lift-off of new wet-ting fronts and an appreciable unsteady rise in the walltemperature.

The authors of the present study observed the samevapor behavior for both near-saturated and subcooled flowboiling in microgravity, Fig. 4 [20]. They also successfullyused the Interfacial Lift-off Model to predict flow-boilingCHF in microgravity for near-saturated flow.

In this study, the Interfacial Lift-off Model is modifiedto predict subcooled flow boiling CHF for both 1 ge andlge conditions. As shown in Fig. 5, the vapor-liquid inter-face is idealized as a sinusoidal wave with amplitude andwavelength increasing in the flow direction. Due to theliquid’s inlet subcooling, bubble formation is expected tooccur some distance from the upstream edge of the heatedwall. However, calculations of the point of bubble depar-ture based on Levy’s model [26] as well as video analysis

ll for near-saturated flow in lge at low and high velocities.

Fig. 5. Idealized wavy vapor–liquid interface with periodic wetting fronts.


showed boiling was initiated within less than 1 mm fromthe leading edge of the heated wall for all conditions ofthe present study. Fig. 5 shows an initial continuousupstream wetting front up to point z0, followed by theformation of the first wetting front at z*.

Like the original model, the present model consists offour sub-models that describe the liquid and vapor flowsand conditions leading to interfacial lift-off. First, a sepa-rated flow model is employed to predict bulk liquidenthalpy and local mean values of liquid velocity, vaporvelocity, and average vapor layer thickness. Second, aninterfacial instability analysis utilizes this information todescribe the spacing of wetting fronts and pressure forcecreated by interfacial curvature. Third, the lift-off criterionis defined as the heat flux required to generate sufficientvapor momentum in the wetting fronts to overcome thepressure force exerted upon the interface due to interfacialcurvature. Fourth, an energy balance is written for theheated wall relating the average surface heat flux at CHFto the heat flux concentrated in the wetting fronts. Addi-tionally, flow visualization plays a key role in model clo-sure by providing support for specification of the ratio ofwetting front length to vapor wavelength.

Unlike the original saturated flow boiling CHF model,the sub-models introduced here are complicated by axialvariations of the liquid bulk enthalpy and partitioning ofwall energy between sensible and latent heat. Followingare detailed derivations for all four sub-models.

3.1. Separated flow model

The present CHF model requires knowledge of bulkliquid enthalpy, liquid phase velocity, vapor phase velocity,and average vapor layer thickness as a function of axial dis-tance. Visual observations at heat fluxes nearing CHF indi-cated that the two-phase mixture was, for the most part,separated over the entire length of the heater [20]. There-fore, the flow is modeled by solving the mass, momentum,and energy conservation equations using a type of sepa-rated two-phase flow model called the slip flow model. Thismodel assumes one-dimensional flow with uniform velocityand enthalpy within each phase, while allowing for differ-ences in the phase velocities. Furthermore, the vapor and

liquid layers are assumed to be in mechanical equilibriumand, hence, pressure is uniform across the flow area atany given axial location. Additional slip flow assumptionsinclude steady state and a uniform wall heat flux. Althoughthe liquid–vapor interface is actually wavy, the separatedflow model is quite effective in providing reliable estimatesof the vapor and liquid velocities and average vapor layerthickness as a function of axial distance.

Fig. 6 provides definitions for the key geometricalparameters used in this model. Liquid is assumed to entera rectangular channel of width W and height H at a massvelocity G. The channel is subjected to a uniform heat fluxq00 along only one side. A vapor layer of thickness d isestablished and grows thicker along the flow direction.Conservation relations are applied below for a channelcontrol volume of length Dz.

The mass flow rate of liquid evaporating at the vapor-liquid interface is expressed by the mass conservation equa-tion as

W 0fg ¼

d

dzðqgU gAgÞ: ð1Þ

Introducing the mass vapor quality yields the followingrelations for the vapor and liquid layers, respectively,

x ¼qgU gAg

GA¼

qgU ga

Gð2Þ

and

1� x ¼ qfU fAf

GA¼ qf U fð1� aÞ

G; ð3Þ

where a ¼ Ag=A ¼ d=H . Combining Eqs. (1) and (2) yields

W 0fg ¼ GA

dxdz: ð4Þ

Applying conservation of momentum to the liquid andvapor control volumes separately results in the followingequations:

G2 d

dzð1� xÞ2

qfð1� aÞ

" #¼ �ð1� aÞ dP

dz�

sw;fpw;f

Aþ sipi

A

� qfð1� aÞga ð5Þ

and

Fig. 6. Separated vapor–liquid flow in rectangular channel subjected to one-sided heating.


G2 d

dzx2

qga

!¼ �a

dPdz�

sw;gpw;g

A� sipi

A� qgaga; ð6Þ

where ga is the component of acceleration acting axiallyopposite to the direction of fluid flow. Summing Eqs. (5)and (6) gives the following expression for pressure gradient,

� dPdz¼ G2 d

dzx2

qgaþ ð1� xÞ2

qfð1� aÞ

$ %þ

sw;gpw;g þ sw;fpw;f

A

þ ½qgaþ qfð1� aÞ�ga: ð7Þ

The wall shear stress in phase k (either liquid or vapor) isgiven by

sw;k ¼fk

2qkU 2

k : ð8Þ

Bhatti and Shah [27] developed the following formula forthe Fanning friction factor applicable to laminar, transi-tion, and turbulent flow regimes,

fk ¼ C1 þC2

Re1=C3D;k

: ð9Þ

For laminar flow ðReD;k 6 2100Þ;C1 ¼ 0;C2 ¼ 16;C3 ¼ 1;for transition flow (2100 < ReD;k 6 4000Þ;C1 ¼ 0:0054;C2 ¼ 2:3� 10�8; C3 ¼ �2=3; and for turbulent flowðReD;k > 4000Þ; C1 ¼ 0:00128; C2 ¼ 0:1143; C3 ¼ 3:2154.The Reynolds number for phase k is based on the hydraulicdiameter,

ReD;k ¼qkU kDk

lk

: ð10Þ

The interfacial shear stress on the liquid by the vapor is de-fined as

si ¼Cf;i

2qgðU g � U fÞ2 ð11Þ

Both Galloway and Mudawar [14] and Gersey and Muda-war [16] critiqued several models for the interfacial frictioncoefficient, Cf ;i, and recommended a constant value of 0.5for a wavy vapor–liquid interface.

The energy conservation equation yields an expressionfor the local bulk liquid enthalpy at any axial location z:

q00ph

GA¼ d

dz½xhg þ ð1� xÞhb�; ð12Þ

Integrating Eq. (12) gives the following expression for bulkliquid enthalpy,

hbðzÞ ¼hi � xðzÞhgðzÞ þ q00ph

GA z

1� xðzÞ ; ð13Þ

If a heat utility ratio given by n is defined as the fraction ofthe surface heat flux that converts stagnant, near-wallliquid at the local bulk temperature to vapor, then

W 0fg½hfgðzÞ þ DhsubðzÞ�dz ¼ nq00ph dz; ð14Þ

where Dhsub is the difference between the saturated enthalpyand local bulk liquid enthalpy. Substitution of Eq. (4) intoEq. (14) yields

dxdz¼ nq00ph

GA½hfgðzÞ þ DhsubðzÞ�: ð15Þ

Assuming the heat utility ratio, n, is known, the mass,momentum, and energy equations can be expressed interms of four independent variables: a (or d), x, hb, andP. All other parameters (e.g. Uf and Ug) can be expressedin terms of these variables. Thermophysical properties inthe liquid are determined by the local bulk liquid tempera-ture and therefore hb. Saturated thermophysical propertiesof the liquid and vapor are determined by the local pres-sure, P. The model equations are solved simultaneouslyto determine the variations of d, x, hb, and P relative toz. The mean liquid velocity, Uf, and mean vapor velocity,Ug, are determined thereafter from these four independent


variables. The heat utility ratio, n, required for this solutionis discussed later.

3.2. Interfacial instability analysis

The periodic, wavy interface between the vapor andliquid prior to CHF resembles the classical case shown inFig. 7a of two fluids moving relative to each other betweentwo infinite parallel walls. Disturbances may lead to anunstable interface and the objective is to find the wave-

Fig. 7. (a) Idealized sinusoidal liquid–vapor interface and (b) vapor moment(adapted from Zhang et al. [20]).

length for which the interface is neutrally stable; that is,on the verge of becoming unstable. An interface having thiscritical wavelength would be most susceptible to themomentum flux of vapor emanating from the wettingfronts. The critical wavelength is obviously a function ofthe mean liquid velocity, Uf, mean vapor velocity, Ug,and the average vapor layer thickness, d, quantities deter-mined from the separated flow model.

Assuming that a disturbance produces a sinusoidalinterface,

um and interfacial pressure difference used to determine lift-off heat flux


gðz; tÞ ¼ g0eikðz�ctÞ; ð16Þ

the pressure rise across the interface can be expressed as

P f � P g ¼ � q00f ðc� U fÞ2 þ q00gðc� U gÞ2 þ ðqf � qgÞgn

k

h ikg;

ð17Þ

where k ¼ 2p=k is the wave number, c is the wave speed,q�f ¼ qf cothðkH fÞ, and q�g ¼ qg cothðkH gÞ. Alternatively, atwo-dimensional interface with small curvature has anapproximate pressure difference of

P f � P g � ro2goz2¼ �rk2g: ð18Þ

Equating pressure difference in Eqs. (17) and (18) yields anexpression for the wave speed:

c ¼q00f U f þ q00gU g

q00f þ q00g

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirk

q00f þ q00g�

q00f q00gðU g � U fÞ2

ðq00f þ q00gÞ2�ðqf � qgÞðq00f þ q00gÞ

gn

k

vuut ; ð19Þ

where gn is the component of acceleration perpendicular tothe heated wall. The first term within the radical in Eq. (19)accounts for the effect of surface tension and is always sta-bilizing. The second term results from the velocity differ-ence between the phases and is always destabilizing. Thelast term can be either stabilizing or destabilizing depend-ing upon the orientation of the interface with respect tothe gravitational field. If the argument within the radicalin Eq. (19) is positive, the wave speed is real, and the inter-face is sinusoidal, stable, and diminishing in amplitude.Allowing for a negative argument results a complex wavespeed, c ¼ cr þ ici. The imaginary component of this wavespeed is

Table 1Present 1ge horizontal flow boiling CHF data

DT sub;o ¼ 3� 2 �C DT sub;o ¼ 10� 2 �C

U (m/s) CHF (W/cm2) U (m/s) CHF (W/cm2)

0.10 27.6 0.5 30.10.19 29.0 0.8 31.30.28 30.4 1.0 31.90.31 31.0 1.2 32.50.46 29.0 1.5 33.10.48 31.30.54 30.80.80 31.00.90 30.10.90 29.21.10 30.01.30 30.91.31 30.21.35 31.31.50 30.3

ci ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq00f q

00gðU g � U fÞ2

ðq00f þ q00gÞ2þðqf � qgÞðq00f þ q00gÞ

gn

k� rk

q00f þ q00g

vuut : ð20Þ

The existence of an imaginary component signifies anunstable interface. Neutral stability occurs when ci ¼ 0.Setting Eq. (20) equal to zero and solving for k yields thefollowing expression for the critical wavelength, kc,

kc ¼2pkcr

¼q00f q

00gðU g � U fÞ2

2rðq00f þ q00gÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq00f q

00gðU g � U fÞ2

2rðq00f þ q00gÞ

" #2

þðqf � qgÞgn

r

vuut : ð21Þ

3.3. Interfacial lift-off criterion

The present model postulates that the liquid–vaporinterface be lifted away from the heated wall when themomentum flux of vapor emanating from the wetting frontexceeds the pressure force acting upon the interface due tothe interface curvature. The heat flux causing interfaciallift-off may be found by equating the average pressure forceacting to maintain interfacial contact and the vapormomentum tending to push the interface away from thesurface as illustrated in Fig. 7b. Integrating Eq. (18) overthe length ‘ ¼ bk of the wetting front and dividing theresult by this length yields the average pressure differencefor the wetting front:

P f � P g ¼4prd

bk2c

sinðpbÞ: ð22Þ

The heat flux, q00w, dedicated to converting a mass of sub-cooled liquid into saturated vapor at a wetting front canbe expressed as

q00wAw ¼ ðhfg þ DhsubÞqgU g;nAw; ð23Þ

DT sub;o ¼ 20� 2 �C DT sub;o ¼ 30� 2�C

U (m/s) CHF (W/cm2) U (m/s) CHF (W/cm2)

0.5 31.6 0.5 34.60.8 33.7 0.8 38.61.0 35.2 1.0 40.91.2 37.4 1.2 43.21.5 40.3 1.5 46.7

Table 2Present lge CHF data

DT sub;o ¼ 4� 2 �C DT sub;o ¼ 8� 2 �C DT sub;o ¼ 32� 2 �C

U

(m/s)CHF(W/cm2)

U

(m/s)CHF(W/cm2)

U

(m/s)CHF(W/cm2)

0.10 10.1 0.10 14.0 0.30 21.20.15 13.8 0.16 13.9 0.50 23.00.19 14.6 0.24 14.5 0.60 24.90.23 14.0 0.29 17.5 0.70 26.00.25 15.4 0.30 18.1 1.10 30.50.26 15.0 0.31 20.4 1.50 35.20.90 23.2 0.46 20.61.10 25.5 0.48 22.31.50 28.7 0.53 22.8

0.80 27.20.90 25.01.30 26.31.31 27.31.35 27.8

Table 3Sturgis and Mudawar [17,18] 1ge horizontal flow boiling CHF data

DT sub;o ¼ 3 �C DT sub;o ¼ 16 �C DT sub;o ¼ 29 �C

U

(m/s)CHF(W/cm2)

U

(m/s)CHF(W/cm2)

U

(m/s)CHF(W/cm2)

0.5 25.1 0.5 35.2 0.5 42.70.5 24.9 0.5 35.5 0.5 43.11.0 28.2 1.0 38.8 1.0 48.81.5 30.5 1.0 38.3 1.0 48.42.0 31.3 1.5 41.8 1.5 54.02.0 32.9 1.5 42.0 1.5 51.53.0 34.3 2.0 44.3 1.5 51.23.0 34.8 2.0 44.0 2.0 59.43.0 34.7 2.0 42.8 3.0 70.74.0 36.8 3.0 47.0 3.0 73.65.0 36.5 4.0 52.4 4.0 86.95.0 36.5 4.0 51.8 5.0 97.16.0 37.8 5.0 63.1 6.0 108.77.0 42.3 6.0 71.7 7.0 118.28.0 46.9 6.0 69.3 8.0 129.3

7.0 76.67.0 76.48.0 85.8

Fig. 8. Comparison of model predictions and present data measured at (a)1 ge and (b) lge.


where Aw is the surface area associated with the wettingfront and U g;n the vapor velocity normal to the surface.

Equating the vapor momentum flux per unit area,qgU 2

g;n, to the average pressure difference, the local lift-offheat flux at the downstream edge of the heat wall can bedetermined as

q00w ¼ qgðhfg þ cp;fDT sub;oÞP f � P g

qg

!1=2

¼ qgðhfg þ cp;fDT sub;oÞ4prd

qgbk2c

sinðpbÞ !1=2

ð24Þ

A statistical examination of interfacial waves by Zhanget al. [19] showed that, just prior to CHF, waves generatedat z� have a tendency to preserve their curvature as theypropagate downstream. CHF then ensues when the most

downstream wetting front begins lifting from the heatedwall near z ¼ L. Therefore d and kc in Eq. (24) are evalu-ated at z�, which is defined as z� ¼ z0 þ kcðz�Þ, where z0 isthe location at which the vapor velocity just exceeds theliquid velocity, both of which are determined from theseparated flow model.

3.4. Energy balance

Zhang et al. [19] showed that the interfacial wavelengthand the wetting front length increase in the flow directionbut the wetting front length remains a constant fractionof the local wavelength with b = 0.20. Prior to CHF, liquidis converted into vapor only in the wetting fronts. The por-tion, n, of the wall heat flux at CHF, q00m, that is dedicatedto vapor generation is related to the localized lift-off heatflux by

nq00m ¼ bq00w: ð25Þ


Combining Eqs. (24) and (25) yields the following expres-sion for CHF,

q00m ¼bn

qgðhfg þ cp;fDT sub;oÞ4prd

qgbk2c

sinðpbÞ! ��

z�

" #1=2

: ð26Þ

4. Model assessment

The present model was developed to predict CHF forflow boiling with different outlet subcoolings at 1 ge andlge. To validate this model, CHF values were measuredfor FC-72 in both ground and parabolic flight experiments.In the ground experiments, the channel was mounted hor-izontally with the heated wall of the channel facing Earth’sgravity (i.e. ga ¼ 0 and gn ¼ 1 geÞ. The measured 1 ge CHFdata are given in Table 1 and include outlet subcooling of3, 10, 20, and 30 �C corresponding to an outlet pressure ofP o ¼ 1:44� 105 Pa (20.9 psi). CHF values measured at lge

(ga ¼ gn ¼ 0) are listed in Table 2 and correspond to thesame outlet pressure as the 1 ge data. Sturgis and Mudawar[17,18] previously performed horizontal flow boiling CHFexperiments with FC-72 at 1 ge in which the heated wallwas vertically mounted (i.e. ga ¼ gn ¼ 0). Their channelhad the same geometry and dimensions as the present studybut a much thicker heated wall. Their data were obtainedat an outlet pressure of 1.38 bars (20 psi) over a velocityrange of 0.25–10 m/s and outlet subcoolings of 3, 16, and29 �C. Listed in Table 3, the Sturgis and Mudawar CHFdata provided added opportunity for validation of thepresent model.

As indicated earlier, solution of the separated flowmodel conservation relations and determination of the

Fig. 9. Comparison of model predictions with 1 ge CH

lift-off heat flux require knowledge of the magnitude ofthe heat utility ratio, n. Information regarding this impor-tant parameter is very hard to track. Rohsenow [28] postu-lated that heat transfer of subcooled flow boiling is thesuperposition of the heat transferred by single-phase con-vection and the heat transferred by bubble nucleation.Levy [29], Gambill [30], Avksentyuk [31], and Siman-Tovet al. [32] assumed that subcooled flow boiling CHF isthe superposition of the heat flux transferred by single-phase forced convection and the heat flux correspondingto subcooled pool boiling. These studies suggest that onlya portion of the wall heat flux is dedicated to vapor gener-ation during subcooled flow boiling even when CHFoccurs.

As discussed earlier, the present study defines the heatutility ratio, n, as the fraction of the wall heat flux that con-verts near-wall liquid at local bulk liquid temperature tovapor. Two key criteria concerning the magnitude of thisparameter are 0 6 n 6 1 for subcooled flow, and n ¼ 1for saturated flow. A dimensionless relation that satisfiesboth criteria was derived from data obtained in the presentstudy as well as those of Sturgis and Mudawar by minimiz-ing least mean absolute error (MAE) in CHF predictionusing the CHF model described above:

n ¼ 1� qf

qg

cpfDT sub;o

hfg

0:00285qfU

2Dr

� �0:2$ %

: ð27Þ

Fig. 8a compares predictions of the CHF model with thepresent 1ge CHF data for four outlet subcoolings. TheCHF model captures all the important data trends, increas-ing CHF with increasing subcooling and/or velocity, and aweakening effect of velocity on CHF with decreased subco-

F data measured by Sturgis and Mudawar [17,18].

Fig. 10. Comparison of model predictions and CHF data measured at1 ge and lge.


oling. The model over-predicts the data for DT sub;o ¼ 20and 30 �C but provides excellent results for DT sub;o ¼ 3and 10 �C. However, good overall predictive ability is evi-denced by the MAE for the individual subcoolings rangingfrom 3.2% to 23.6%. Notice that the model predictions ex-clude very low velocities where flow boiling resembles poolboiling rather than the wavy vapor regime upon which thepresent model is based.

Fig. 8b shows a similar comparison of the model predic-tions with the present microgravity data for three outletsubcoolings. The model over-predicts the CHF data forDT sub;o ¼ 32 �C but provides very good predictions forDT sub;o ¼ 4 and 8 �C. However, good overall predictivecapability is shown for these lge data as well with theMAE for individual subcoolings ranging from 9.9% to18.1%.

Fig. 9 shows the model accurately predicts the Strugisand Mudawar 1 ge CHF data for all velocities and ouletsubcoolings. The MAE of the model predictions is 7.7%,8.8% and 4.6% for DT sub;o ¼ 3, 16 and 29 �C, respectively.

To further illustrate both the accuracy and the versatil-ity of the present model at predicting CHF data for both1 ge and lge and both near-saturated and subcooled flow,the three afore-mentioned databases are compared to themodel predictions in Fig. 10. Virtually all data fall within±25% of the model predictions and the MAE for the com-bined database is 9.4%.

5. Conclusions

This study extended the Interfacial Lift-off Model orig-inally derived for saturated flow boiling CHF to subcooledflow boiling. New CHF data were measured in 1 ge groundtests and lge tests performed in parabolic flight trajectory.

The new database included broad variations of both flowvelocity and outlet subcooling and served to validate theextended CHF model. Key findings from this study areas follows:

(1) Experimental methods, especially heated wall design,were developed that yielded reliable subcooled flowboiling CHF data in both Earth gravity and micro-gravity. These experimental methods also featuredhigh-speed video imaging and analysis of theliquid–vapor interface during the CHF transient.Both the CHF data and the video records played avital role in constructing and validating the extendedCHF model.

(2) A heat utility ratio function was correlated from dataobtained in the present and earlier studies to accountfor the partitioning of wall energy between sensibleand latent heat. This function enables the applicationof the Interfacial Lift-of Model to subcooled flowboiling at both lge and 1 ge.

(3) The new extended Interfacial Lift-off Model is veryeffective at predicting both near-saturated and sub-cooled flow boiling CHF at 1 ge and lge.

(4) Future studies should address conditions that yield asubstantial inlet non-boiling region. Different pointof net vapor generation models could be easily incor-porated into the present CHF model to extend itsapplicability to other fluids and operating conditions.

Acknowledgements

The authors are grateful for the support of the NationalAeronautics and Space Administration under Grant No.NNC04GA54G. Several individuals provided valuableassistance in preparing the apparatus for the flight experi-ments and obtaining flight data. The authors especiallythank John McQuillen, Kirk Logsoon and James Withrowof the NASA Glenn Center for their assistance.

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